Can a GIS 'store' non-spatial data. Well, yes, attribute data can be stored in a GIS, at the end of the day, its just a relational database.....
Can a GIS store a model file? Again in theory, GIS is just a relational database, and if you want to use the GIS database to store content, in theory this would be possible.
Thirdly " is there a way to seamlessly convert the files between systems? "
So this is a big jump in capability when compared to your other questions. The short answer to this is probably no, not in the way you think it would.
Simple scenarios like DGN/DWG/DXF files, would need to be drawn in very standard ways (line style, colour, level/layer names etc etc) and then - Most GIS systems would be able to be configured to open these and display within their own system, either desktop or web. I would include your flood model scenario into this statement (again, you would need standardised CAD/engineering layers. eg: Your flood model is saved as a raster based grid file format. GIS should natively open this.
Once you head outside of that though, complex engineering models (BIM), or mining models for example, there are too many variables and unknowns and in general, file formats which are not supported 'natively'. 'Middleware' as you state it may provide more of an ETL solution (ie: Translate out of BIM model into GIS data) but that moves away completely from your original 'single source of truth' concept.
My additional comment would be that - engineer's tend to want to work in a BIM/CAD space more than a GIS space, and CAD software has come a long way from providing visualisation tools that now rival GIS visualisations, so at the moment, it may be easier to go from GIS - BIM both from technical and end-user perspective.